Integrand size = 25, antiderivative size = 65 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a x}{2}+\frac {a \cos (c+d x)}{d}+\frac {a \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2917, 2670, 14, 2671, 294, 327, 209} \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {a \cos (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}+\frac {a \sec (c+d x)}{d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac {3 a x}{2} \]
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Rule 14
Rule 209
Rule 294
Rule 327
Rule 2670
Rule 2671
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \sin (c+d x) \tan ^2(c+d x) \, dx+a \int \sin ^2(c+d x) \tan ^2(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac {a \text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {a \cos (c+d x)}{d}+\frac {a \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {3 a x}{2}+\frac {a \cos (c+d x)}{d}+\frac {a \sec (c+d x)}{d}+\frac {3 a \tan (c+d x)}{2 d}-\frac {a \sin ^2(c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 a (c+d x)}{2 d}+\frac {a \cos (c+d x)}{d}+\frac {a \sec (c+d x)}{d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(\frac {a \left (-12 d x \cos \left (d x +c \right )+\sin \left (3 d x +3 c \right )+4 \cos \left (2 d x +2 c \right )+9 \sin \left (d x +c \right )+16 \cos \left (d x +c \right )+12\right )}{8 d \cos \left (d x +c \right )}\) | \(63\) |
risch | \(-\frac {3 a x}{2}+\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 a}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(71\) |
derivativedivides | \(\frac {a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(94\) |
default | \(\frac {a \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d}\) | \(94\) |
norman | \(\frac {\frac {3 a x}{2}-\frac {4 a}{d}-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {4 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(154\) |
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Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {a \cos \left (d x + c\right )^{3} - 3 \, a d x + 2 \, a \cos \left (d x + c\right )^{2} - 3 \, {\left (a d x - a\right )} \cos \left (d x + c\right ) + {\left (3 \, a d x + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=a \left (\int \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {{\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a - 2 \, a {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{2 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {3 \, {\left (d x + c\right )} a + \frac {4 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 12.68 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.46 \[ \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {\left (\frac {a\,\left (3\,d\,x-6\right )}{2}-\frac {3\,a\,d\,x}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (3\,a\,d\,x-\frac {a\,\left (6\,d\,x-6\right )}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {a\,\left (6\,d\,x-10\right )}{2}-3\,a\,d\,x\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {3\,a\,d\,x}{2}-\frac {a\,\left (3\,d\,x-2\right )}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,\left (3\,d\,x-8\right )}{2}-\frac {3\,a\,d\,x}{2}}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {3\,a\,x}{2} \]
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